split plot in time and space and combined analyses€¦ · split plot in time and the nested block...

SPLIT PLOT IN TIME AND THE NESTED BLOCK ARRANGEMENTS

Split Plot in Time

This arrangement occurs when you have an experiment where you collect data from

the same experimental unit over a series of dates.

An example of this would be an experiment that includes a perennial species (e.g.

alfalfa) and different harvest dates or an experiment where you measure water use

each week in a field experiment.

The ANOVA for this situation is similar to a split block.

Example

An experiment comparing the yield of 10 alfalfa cultivars cut at three different times

during the growing season. Cultivars and cutting times are fixed effects.

SOV Df Expected Mean Squares F-test

Rep r-1

2222

Rcttc

Cultivar

c-1

1

2

22

c

rtt

C MS/Error(a) MS

Error(a) (r-1)(c-

1)

22

t

Time

t-1

1

2

22

trcc

T MS/Error(b) MS

Error(b) (r-1)(t-1)

22

c

Cultivar*Time

(c-1)(t-1)

)1)(1(

)( 2

2

tc

r

C*T MS/Error(c) MS

Error(c) (r-1)(c-

1)(t-1)

2

Total rct-1

LSD’s for Split Plot in Time Arrangement

1. To compare two A means averaged over all time treatments (.e.g. a0 vs a1)

ta /2, Error(a) df

2Error(a)MS

r * time

= 4.3032(2.5150)

3X4

= 2.79

2. To compare two time means averaged over all A treatments (.e.g. time0 vs time1)

59.1

23

)412.0(2303.4

)(2t df Error(b) /2,

X

ra

MSbError

3. To compare two levels of A means at the same level of the time factor (e.g.

a0time0 vs a1time0)

072.3

43

]515.2794.0)14[(23.4LSD and

4.3515.2794.0)14(

)303.4(515.2)447.2)(794.0)(14(

)()()1(

)()()()()1(

*

])()()1[(2t

'

)(,2/)(,2/

'

'

x

t

MSaErrorMScErrortime

tMSaErrortMScErrortimet

and

timer

MSaErrorMScErrortime

ac

dfaErrordfcError

ac

ac

4. To compare two time means at the same level of factor A (e.g. a0time0 vs a0time1)

551.1

23

]412.0794.0)12[(2447.2LSD and

447.2412.0447.2)12(

)447.2(412.0)447.2)(794.0)(12(

)()()1(

)()()()()1(

])()()1[(2t

'

)(,2/)(,2/

'

'

x

t

MSbErrorMScErrora

tMSbErrortMScErrorat

and

ra

MSbErrorMScErrora

bc

dfbErrordfcError

bc

bc

5. To compare two vertical means at different levels of the horizontal factor (e.g. a0b0 vs

a1b3)

294.2

423

]515.2)12(412.0)14(794.0)14)(12[(2208.3LSD and

208.31515.2)12(412.0)14(447.2)14)(12(

515.2)12()447.2(412.0)14()447.2)(794.0)(14)(12()

)()1()()1()()1)(1(

)()()1()()()1()()()1)(1(

**

])()1()()1()()1)(1[(2t

'

)(,2/)(,2/)(,2/

'

'

xx

t

MSaErroraMSbErrortimeMScErrortimea

tMSaErroratMSbErrortimetMScErrortimeat

and

timear

MSaErroraMSbErrortimeMScErrortimea

abc

dfaErrordfbErrordfcError

abc

abc

SAS Commands for an RCBD with a Split Plot in Time Arrangement

options pageno=1;

data spplttim;

input A $ time $ Rep Trt Yield;

datalines;

a0 b0 1 1 13.8

a0 b1 1 2 15.5

a0 b2 1 3 21

a0 b3 1 4 18.9

a1 b0 1 5 19.3

a1 b1 1 6 22.2

a1 b2 1 7 25.3

a1 b3 1 8 25.9

a0 b0 2 1 13.5

a0 b1 2 2 15

a0 b2 2 3 22.7

a0 b3 2 4 18.3

a1 b0 2 5 18

a1 b1 2 6 24.2

a1 b2 2 7 24.8

a1 b3 2 8 26.7

a0 b0 3 1 13.2

a0 b1 3 2 15.2

a0 b2 3 3 22.3

a0 b3 3 4 19.6

a1 b0 3 5 20.5

a1 b1 3 6 25.4

a1 b2 3 7 28.4

a1 b3 3 8 27.6

;;

ods rtf file='example.rtf';

run;

proc anova;

class rep a time;

model yield=rep a rep*a time rep*time a*time;

test h=a e=rep*a;

test h=time e=rep*time;

means a/lsd e=rep*a;

means time/lsd e=rep*time;

means a*time;

title 'ANOVA for an RCBD with a Split Plot in Time

Arrangement';

run;

ods rtf close;

run;

ANOVA for an RCBD with a Split Split Plot in Time Arrangement

The ANOVA Procedure

Class Level Information

Class Levels Values

Rep 3 1 2 3

A 2 a0 a1

time 4 b0 b1 b2 b3

Number of Observations Read 24

Number of Observations Used 24


The ANOVA Procedure

Dependent Variable: Yield

02:15 Sunday, June 17, 2012 205

Source DF

Sum of

Squares Mean Square F Value Pr > F

Model 17 511.3587500 30.0799265 37.91 0.0001

Error 6 4.7608333 0.7934722

Corrected Total 23 516.1195833

R-Square Coeff Var Root MSE Yield Mean

0.990776 4.298913 0.890771 20.72083

Source DF Anova SS Mean Square F Value Pr > F

Rep 2 7.8658333 3.9329167 4.96 0.0536

A 1 262.0204167 262.0204167 330.22 <.0001

Rep*A 2 5.0358333 2.5179167 3.17 0.1148

time 3 215.2612500 71.7537500 90.43 <.0001

Rep*time 6 2.4775000 0.4129167 0.52 0.7767

A*time 3 18.6979167 6.2326389 7.85 0.0168

Tests of Hypotheses Using the Anova MS for Rep*A as an Error Term


A 1 262.0204167 262.0204167 104.06 0.0095

Tests of Hypotheses Using the Anova MS for Rep*time as an Error Term


time 3 215.2612500 71.7537500 173.77 <.0001


The ANOVA Procedure

02:15 Sunday, June 17, 2012 206


The ANOVA Procedure

t Tests (LSD) for Yield

02:15 Sunday, June 17, 2012 207

Note

:

This test controls the Type I comparisonwise error rate, not the

experimentwise error rate.

Alpha 0.05

Error Degrees of Freedom 2

Error Mean Square 2.517917

Critical Value of t 4.30265

Least Significant Difference 2.7873

Means with the same letter are

not significantly different.

t Grouping Mean N A

A 24.0250 12 a1

B 17.4167 12 a0


The ANOVA Procedure

02:15 Sunday, June 17, 2012 208


The ANOVA Procedure


02:15 Sunday, June 17, 2012 209

Note

:

This test controls the Type I comparisonwise error rate, not the

experimentwise error rate.

Alpha 0.05

Error Degrees of Freedom 6

Error Mean Square 0.412917

Critical Value of t 2.44691

Least Significant Difference 0.9078

Means with the same letter are

not significantly different.

t Grouping Mean N time

A 24.0833 6 b2

B 22.8333 6 b3

C 19.5833 6 b1

D 16.3833 6 b0


The ANOVA Procedure


02:15 Sunday, June 17, 2012 204

Level of

A

Level of

time N

Yield

Mean Std Dev

a0 b0 3 13.5000000 0.30000000

a0 b1 3 15.2333333 0.25166115

a0 b2 3 22.0000000 0.88881944

a0 b3 3 18.9333333 0.65064071

a1 b0 3 19.2666667 1.25033329

a1 b1 3 23.9333333 1.61658075

a1 b2 3 26.1666667 1.95021366

a1 b3 3 26.7333333 0.85049005

Randomized Nested Block Arrangement

In some multifactor arrangements, the levels of one factor (e.g., factor B) are similar

but not identical for different levels of another factor (e.g., factor A).

An example would be the agronomic evaluation of genotypes with and without

resistance to a non-selective herbicide such as glyphosate or glufosinate in a single

experiment.

If you use a split plot arrangement with herbicide as the whole plot and genotype as

the subplot, all genotypes would get treated with the herbicide regardless if they are

resistant or non-resistant to the herbicide.

Example

Whole = herbicide rate (0 and X-rate)

Subplot = genotype (Resistant 1, Resistant 2, Susceptible 1 and Susceptible 2

X-rate 0-rate

Resistant 1 Susceptible 2

Resistant 2 Resistant 2

Susceptible 1 Susceptible 1

Susceptible 2 Resistant 1

ANOVA

Sources of variation degrees of freedom

Replicate r-1

Herbicide rate h-1

Error(a) (r-1)(h-1)

Genotype (g-1)

Herbicide rate x Genotype (h-1)(g-1)

Error(b) by subtraction

Total (r x h x g)-1

What are the consequences of using this arrangement?

An alternative to the split plot arrangement would be to block such that only the

resistant genotypes get sprayed with herbicide and the susceptible genotypes do not

get sprayed with herbicide.

X-rate 0-rate



This arrangement allows for valid comparisons between:

1. Genotypes nested within an herbicide level (e.g. Resistant 1 vs. Resistant 2

or Susceptible 1 vs. Susceptible 2).

2. Herbicide rate averaged across genotypes (e.g. X-rate vs. 0-rate).

Valid comparisons cannot be made between genotypes nested in different

herbicide rates (e.g. X-rate and Resistant 1 vs. 0-rate and Susceptible 2).

ANOVA

Sources of variation Degrees of

freedom†

Method for calculating the F-statistic

Replicate r-1 Replicate MS/Error MS

Herbicide h-1 Herbicide MS/Error MS

Genotype(Herbicide) (b-1)+(nb-1) Genotype(Herbicide MS/Error MS

Error (r-1)(G-1)

Total (r x G)-1

†r = number of replicates, h = number of herbicide treatments, b = number of biotech

(i.e., resistant) genotypes, nb = number of non-biotech (i.e., susceptible) genotypes, and

G = total number of genotypes in the experiment (i.e., b + nb).

Mean Separation

Valid mean separation can be done between:

1. Genotypes within a herbicide level (e.g. Resistant 1 vs. Resistant 2 or

Susceptible 1 vs. Susceptible 2).

The formula for calculating the least significant difference (LSD)

between genotypes within a class at p=0.05 is:

r

MSError x 2dfError ,2/05.0t

2. Herbicide rates averaged across genotypes (e.g., X-rate vs. 0-rate).

The formula for calculating the least significant difference (LSD)

between herbicide rates averaged across genotypes at p=0.05 is:

nbr x

1

br x

1MSError dfError ,2/05.0t

Where b = number of biotech (i.e., resistant) genotypes and

nb = number of non-biotech (i.e., susceptible) genotypes.

split plot in time and space and combined analyses€¦ · split plot in time and the nested block...

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